The Red Devils recovered from an early scare to beat Eddie Howe’s men thanks to goals from Anthony Martial, Marcus Rashford, Fernandes and Greenwood.
Junior Stanislas gave the visitors a surprising lead 15 minutes into the game, but the hosts pulled parity on the 29th minute mark when a Fernandes pass found Greenwood, who smacked a clinical low strike across Aaron Ramsdale and into the back of the net.
Rashford gave Ole Gunnar Solskjaer’s men the lead from the penalty spot moments later after Mike Dean awarded a penalty for Adam Smith’s handball.
Martial doubled United’s lead on the cusp of halftime after he cut in from the left before bending a brilliant effort into the top corner.
The Cherries were given a penalty five minutes after the break, which Josh King slotted home to set up a whirlwind start to the second-half at Old Trafford.
Eddie Howe’s men thought they had gotten the equalizer when Arnaut Danjuma blasted a shot past David De Gea, but his effort was flagged offside.
Greenwood scored his second goal of the game with a right-footed shot before Fernandes, who provided two assists earlier, bent an unstoppable free kick into the back of the net.
The win has catapulted United firmly into the Champions League qualification picture and they now sit one point above Chelsea, who entertain Watford at Stamford Bridge later in the day.
The Manchester-based club are now unbeaten in 16 games in all competitions; the best current run among all Premier League sides.
